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• CommentRowNumber1.
• CommentAuthorRodMcGuire
• CommentTimeNov 4th 2013

I’ve added the following definition to power set

• the slice category $Inj/S$, where Inj is the wide subcategory of Set with morphisms restricted to injections. This is similar to the subobject definition but is more unpacked. $Inj/S$ has objects that are injections to $S$ and morphisms that are commuting triangles of injections.

I’ve seen $Inj$ appear in discussions (as as a simple thing everybody knows) but sometimes there iis confusion about its properties. Could it use its own page?

(as usual I may be confused/misguided here)

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeNov 5th 2013

I would say feel free to create a page Inj.

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeNov 5th 2013

The only thing I’ll say for now is that I’m not convinced $Inj$ is the best notation. It seems more usual practice to name a category after its objects than after its morphisms; for this reason, instead of $Inj$, I might consider $Set_{inj}$. (Elsewhere we do refer to a category $\mathcal{M}$ who objects are monomorphisms or injections between sets and whose morphisms are commutative squares, so while $Inj$ is untaken, I’m just slightly leery about it.)

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeNov 5th 2013

Todd, I considered making the same comment. Unfortunately, $Set_{inj}$ is difficult to use as a name for an nLab page… Maybe this should just go on the page injection?

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeNov 5th 2013

Yeah, a dedicated section on this category at injection might not be a bad idea. Finite sets and injections is also a category of note that could be written about (semicartesian operads, Schanuel topos, etc.) – or maybe we already have something on this?

• CommentRowNumber6.
• CommentAuthorRodMcGuire
• CommentTimeNov 5th 2013

I would prefer to have a separate page for $Inj$ that also describes $FinInj$. Of course the page should actually be named something like “category of injections between sets” and various alternative abbreviations of the category names can be given there.

Having such a page would help find where these categories are used in the nLab if things are linked correctly. It is rather hard to find mention of these categories by Googling for such common words as “category” and “injection”.

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeNov 5th 2013

Okay, Rod, that sounds good to me.

• CommentRowNumber8.
• CommentAuthorTobyBartels
• CommentTimeNov 25th 2013

Was anything created?

• CommentRowNumber9.
• CommentAuthorRodMcGuire
• CommentTimeSep 5th 2017